On metachronism in ciliary systems: A model describing the dependence of the metachronal wave properties on the intrinsic ciliary parameters

Abstract

A mathematical model is proposed to explain the dependence of the direction and the length of the metachronal wave on parameters that characterize the ciliary beat, the dimensions of the cilia, and the geometry of their arrangement on the ciliated surface. The metachronal wave is decomposed into two mutually perpendicular components, which are chosen in such a way that the direction of one of them is in the direction of the effective stroke. The magnitudes of the two components are determined by using the concept of the time of delay between adjacent cilia. The properties of the metachronal wave are then calculated as a function of the ciliary parameters. The results obtained with the present model predict that the direction of the wave propagation is strongly dependent on the type of metachronism in the direction of the effective stoke and the polarization in time and in space of the ciliary beat. The metachronal wavelength is found to depend on four parameters: the ciliary length, the angle of the arc projected on the cell surface by the ciliary tip during the recovery stroke, the degree of asymmetry of ciliary beat, and the portion of the cycle occupied by the pause. The metachronal wavelength is also found to be only weakly dependent on the ciliary frequency. At this stage there exists relatively little experimental information with which to characterize fully the metachronal properties of ciliary systems. Even when only partial information exists, the model allows prediction, to within a certain range, of the direction of the wave propagation. It also suggests a possible mechanism for the influence of changes in environmental conditions on wave direction and wavelength. In several cases in which full information does exist, good agreement between the experimental findings and the predictions of the model is found. According to this model it will be worthwhile to invest more effort in measuring the time and space polarization of ciliary beating and times of delay between cilia.